Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation

Author:
G. Toscani

Journal:
Quart. Appl. Math. **57** (1999), 521-541

MSC:
Primary 82C31; Secondary 35Q99

DOI:
https://doi.org/10.1090/qam/1704435

MathSciNet review:
MR1704435

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Abstract: We reckon the rate of exponential convergence to equilibrium both in relative entropy and in relative Fisher information, for the solution to the spatially homogeneous Fokker-Planck equation. The result follows by lower bounds of the entropy production which are explicitly computable. Second, we show that the Gross's logarithmic Sobolev inequality is a direct consequence of the lower bound for the entropy production relative to Fisher information. The entropy production arguments are finally applied to reckon the rate of convergence of the solution to the heat equation towards the fundamental one in various norms.

**[Ar88]**L. Arkeryd,*Stability in for the spatially homogeneous Boltzmann equation*, Arch. Rat. Mech. Anal.**103**, 151-167 (1988) MR**946973****[BE85]**D. Bakry and M. Emery,*Diffusions hypercontractives*, Seminaire de Probabilités XIX, Lecture Notes in Math., Vol. 1123, Springer-Verlag, Berlin, 1985, pp. 179-206 MR**889476****[B165]**N. M. Blachman,*The convolution inequality for entropy powers*, IEEE Trans. Info. Theory**2**, 267-271 (1965) MR**0188004****[BGK54]**P. L. Bhatnagar, E. P. Gross, and M. Krook,*A model for collision processes in gases*I.*Small amplitude processes in charged and neutral one-component systems*, Phys. Rev.**94**, 511-525 (1954)**[BT92]**A. V. Bobylev and G. Toscani,*On the generalization of the Boltzmann H-theorem for a spatially homogeneous Maxwell gas*, J. Math. Phys.**33**, 2578-2586 (1992) MR**1167961****[Ca91]**E. A. Carlen,*Superadditivity of Fisher's information and logarithmic Sobolev inequalities*, J. Funct. Anal.**101**, 194-211 (1991) MR**1132315****[CC92]**E. A. Carlen and M. C. Carvalho,*Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation*, J. Stat. Phys.**67**, 575-608 (1992) MR**1171145****[CC94]**E. A. Carlen and M. C. Carvalho,*Entropy production estimates for Boltzmann equations with physically realistic collision kernels*, J. Stat. Phys.**74**, 743-782 (1994) MR**1263387****[CGT96]**E. A. Carlen, E. Gabetta, and G. Toscani,*Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas*, preprint (1996) MR**1669689****[CS91]**E. A. Carlen and A. Soffer,*Entropy production by block variable summation and central limit theorem*, Commun. Math. Phys.**140**, 339-371 (1991) MR**1124273****[Ce88]**C. Cercignani,*Theory and applications of the Boltzmann equation*, Springer-Verlag, New York, 1988 MR**1313028****[Ch43]**S. Chandrasekhar,*Stochastic problems in physics and astronomy*, Rev. Modern Phys.**15**, 1-110 (1943) MR**0008130****[ChC58]**S. Chapman and T. G. Cowling,*The Mathematical Theory of Non-Uniform Gases*, Cambridge University Press, Cambridge, England, 1958**[Cs62]**I. Csiszar,*Informationstheoretische Konvergenzbegriffe im Raum der Wahrscheinlichkeitsverteilungen*, Publications of the Mathematical Institute, Hungarian Academy of Sciences, VII, Series A, 137-158 (1962)**[De89]**L. Desvillettes,*Entropy dissipation rate and convergence in kinetic equations*, Commun. Math. Phys.**123**, 687-702 (1989) MR**1006301****[Fi25]**R. A. Fisher,*Theory of statistical estimation*, Proc. Cambridge Philos. Soc.**22**, 700-725 (1925)**[FHL60]**H. L. Frisch, E. Helfand, and J. L. Lebowitz,*Nonequilibrium distribution functions in a fluid*, Phys. of Fluids**3**, 325-338 (1960) MR**0114351****[Ga95]**E. Gabetta,*On a conjecture of McKean with application to Kac's model*, Special issue devoted to the Proceedings of the 13th International Conference on Transport Theory (Riccione, 1993), Transport Theory Statist. Phys.**24**, 305-317 (1995) MR**1316923****[GT94]**E. Gabetta and G. Toscani,*On convergence to equilibrium for Kac's caricature of a Maxwell gas*, J. Math. Physics**35**, 190-208 (1994) MR**1252106****[GTW95]**E. Gabetta, G. Toscani, and B. Wennberg,*Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation*, J. Statist. Phys.**81**, 901-934 (1995) MR**1361302****[Gr75]**L. Gross,*Logarithmic Sobolev inequalities*, Amer. J. Math.**97**, 1061-1083 (1975) MR**0420249****[Ku67]**S. Kullback,*A lower bound for discrimination information in terms of variation*, IEEE Trans. Information Theory**4**, 126-127 (1967)**[LT95]**P. L. Lions and G. Toscani,*A strengthened central limit theorem for smooth densities*, J. Funct. Anal.**128**, 148-167 (1995) MR**1322646****[Mk66]**H. P. McKean, Jr.,*Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas*, Arch. Rational Mech. Anal.**21**, 343-367 (1966) MR**0214112****[Na58]**J. Nash,*Continuity of solutions of parabolic and elliptic equations*, Amer. J. Math.**80**, 931-954 (1958) MR**0100158****[Ri84]**H. Risken, The Fokker-Planck Equation, Methods of Solution and Applications, Springer-Verlag, Berlin, 1984 MR**749386****[St59]**A. J. Stam,*Some inequalities satisfied by the quantities of information of Fisher and Shannon*, Information and Control**2**, 101-112 (1959) MR**0109101****[To96]**G. Toscani,*Kinetic approach to the asymptotic behaviour of the solution to diffusion equations*, Rendic. di Matem., Serie VII**16**, 329-346 (1996) MR**1404498****[TM80]**C. Truesdell and R. G. Muncaster,*Fundamentals of Maxwell's kinetic theory of a simple monoatomic gas*, Academic Press, New York, 1980 MR**554086****[We93]**B. Wennberg,*Stability and exponential convergence for the Boltzmann equation*, Thesis, Chalmers University of Technology, 1993

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DOI:
https://doi.org/10.1090/qam/1704435

Article copyright:
© Copyright 1999
American Mathematical Society